Octahedron - Area and Volume

Area and Volume

The surface area A and the volume V of a regular octahedron of edge length a are:

Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).

If an octahedron has been stretched so that it obeys the equation:

The formula for the surface area and volume expand to become:

Additionally the inertia tensor of the stretched octahedron is:

I = \begin{bmatrix} \frac{1}{10} m (y_m^2+z_m^2) & 0 & 0 \\ 0 & \frac{1}{10} m (x_m^2+z_m^2) & 0 \\ 0 & 0 & \frac{1}{10} m (x_m^2+y_m^2) \end{bmatrix}

These reduce to the equations for the regular octahedron when:

Read more about this topic:  Octahedron

Famous quotes containing the words area and/or volume:

    “Self-esteem is the real magic wand that can form a child’s future. A child’s self-esteem affects every area of her existence, from friends she chooses, to how well she does academically in school, to what kind of job she gets, to even the person she chooses to marry.”
    —Stephanie Martson (20th century)

    “The other 1000 are principally the ‘old Yankee stock,’ who have lost the town, politically, to the Portuguese; who deplore the influx of the ‘off-Cape furriners’; and to whom a volume of genealogy is a piece of escape literature.”
    —For the State of Massachusetts, U.S. public relief program (1935-1943)